Theory of Bohr s model of atom

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1 Theory of Bohr s model of atom Bohr developed his atomic model in 1911 while working in Manchester, UK with Rutherford, who proposed nuclear theory of atomic structure from his work on scattering of alpha particles. In 1913, Bohr refined his atomic theory by applying quantum mechanics to explain the line spectrum of emitted radiations by atoms. Max Planck proposed that radiation emitted or absorbed by a perfect black body should always be in discrete amounts or quanta. Bohr was the first to apply the new quantum mechanics to the atom. His theory of electrons moving in well-defined orbits around a nucleus containing the protons and neutrons is called the "planetary" model. Central to his model were two postulates: Postulate (1) In the absence of radiation absorption or emission, electrons stay in a stationary state. Postulate (2) Absorption occurs only in discrete amounts, corresponding to a change in energy between two stationary states of the electron. ΔE = hν then gives the frequency of the radiation. Assume an electron in an initial stable state with a larger energy, E nh undergoes a transition to a final stationary state of a lower energy, E nl then according to Einstein, energy of a radiated quantum or photon must be equal to the energy difference, ΔE, between the two stable states, ΔE = E nh - E nl = hυ, where h is called Plank s constant and υ is the frequency of emitted or absorbed electromagnetic radiation. According to dual nature of light, the frequency υ of emitted or absorbed photon is related to its wavelength, λ = υ / c, where c = m/s is the speed of light in free space. Let us now apply these quantum ideas to the circular electron motion. Assume an electron of mass m e, charge e, velocity v n orbits in a circular orbit of radius r n around the nucleus of charge Ze. Total energy of the electron E n is sum of its kinetic energy, KE and potential energy, PE: 1

2 shows that total energy of the electron is negative, indicating that it should confine to a stable orbit of the nucleus. From classical mechanics: According to Bohr s postulate, the magnitude of quantized angular momentum, L n, of an electron in a stationary state is where r n is the radius of n th orbit and an allowed value of principal quantum number is n = 1, 2, 3, 4,, i.e., it can take only integral values in accordance with quantum theory. From equations 2 and 3 : From substituted equation 4 in equation 2 get: For H atom n=1 and Z=1 so: r 1 =0.529A 0 2

3 For ions like H such as He +1.Li +2 and Be +3 alls have 1S 1 One may insert r n from Eq. (5) in Eq. (1) to determine the energy of an electron in any stationary energy state Assume n L is quantum number of an electron in its initial or ground state and n H is that for a higher or an excited state of an atom, we may write Eq. (6) in a more useful fashion: Some references expresses of equation above in formula: Eq.7 can be used to compute the transition energy when an electron jumps from one state to another depending on their lower and higher quantum numbers, n L and n H respectively. Since: ΔE = h υ υ = ΔE / h 3

4 Since: υ =c/ λ υ = ῡ.c ῡ= υ/ c R H = Rydberg constant for hydrogen = X l0 7 m -1 =1.097 X l0 5 cm -1 Since: ΔE = h c ῡ i.e. or 4

5 Atomic spectrum of Hydrogen White light is a combination of light of many different wavelengths. When passed through a prism, white light is spread into its constituent wavelengths, resulting in a band spectrum. A band spectrum resembles a rainbow and contains many different wavelengths of light. When light from a gas discharge tube is passed through a prism, the result is a line spectrum. In contrast to a band spectrum, a line spectrum contains only certain discreet wavelengths of light. Each element gives a characteristic line spectrum, the lines arising from electron transitions within the atom. With one electron and one proton, hydrogen is the simplest element and gives the simplest line spectrum. For example, there are only four lines in the visible region of the hydrogen spectrum, at 656 nm (red), 486 mm (blue-green), 434 nm (blue-violet), and 410 nm (violet). There are actually three series of lines in the hydrogen spectrum, one in the infrared region, one in the visible region, and one in the ultraviolet region. The ultraviolet series involves transitions to the first energy level (n = 1) of the hydrogen atom, the visible series involves transitions to the second energy level (n = 2), and the infrared series involves transitions to the third energy level. The energy of electromagnetic radiation is indirectly related to wavelength, meaning that the longer the wavelength the lower the energy. Therefore, given the four visible lines above, the red line at 656 nm is the longest wavelength and must correspond to the lowest energy transition (3 2). Similarly, the blue-green line at 486 nm corresponds to the next lowest energy transition (4 2), and so on. 5

6 The energy of any level in Hydrogen atom determined by equation: The lowest energy state is called the ground state, this corresponds to n = 1, energy is 13.6 ev. The next energy level has an energy of 3.40 ev.the energies can be compiled in an energy level diagram. The ionization energy is the energy needed to completely remove the electron from the atom. The ionization energy for hydrogen is 13.6 ev. A more generalized equation can be used to find the wavelengths of any spectral lines For the Lyman series, n L = 1 For the Balmer series, n L = 2 For the Paschen series, n L = 3 n H = 2, 3, 4,.... n H = 3, 4, 5.. n H = 4, 5,6.. 6

7 For the Brackett series, n L = 4 n H = 5,6,7.. Whenever an transition occurs between a state, n H to another state, n L (where n H > n L ), a photon is emitted. The photon has a frequency υ = (E H E L )/h and wavelength λ Example: At what wavelength will emission from be observed? n = 4 to n = 1 for the H atom Some successes of the Bohr Theory Explained several features of the hydrogen spectrum Accounts for Balmer and other series Predicts a value for R H that agrees with the experimental value Gives an expression for the radius of the atom Predicts energy levels of hydrogen Can be extended to hydrogen-like atoms, those with one electron 7

8 Modifications of the Bohr Theory 1- Elliptical Orbits Sommerfeld extended the results to include elliptical orbits. Retained the principle quantum number, n Added the orbital quantum number, l l ranges from 0 to n-1 in integer steps All states with the same principle quantum number are said to form a shell The states with given values of n and l are said to form a subshell Electrons are arranged in shells and subshells of orbitals. n shell l subshell m l designates an orbital within a subshell 2- Zeeman Effect Another modification was needed to account for the Zeeman effect.the Zeeman effect is the splitting of spectral lines in a strong magnetic field H. This indicates that the energy of an electron is slightly modified when the atom is immersed in a magnetic field. A new quantum number, m l, called the orbital magnetic quantum number, had to be introduced. m l can vary from - l to + l in integer steps. Total of orbitals in l th subshell = 2 l + 1 l m l , 0, , -1, 0, +1, +2 8

9 3-Fine Structure High resolution spectrometers show that spectral lines are, in fact, two very closely spaced lines, even in the absence of a magnetic field. This splitting is called fine structure. Another quantum number, m s, called the spin magnetic quantum number, was introduced to explain the fine structure. m s can take values m s =±1/2. 9

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